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Batch Renewal Arrival Process Subject to Geometric Catastrophes

Author

Listed:
  • F. P. Barbhuiya

    (Indian Institute of Technology Kharagpur)

  • Nitin Kumar

    (Indian Institute of Technology Kharagpur)

  • U. C. Gupta

    (Indian Institute of Technology Kharagpur)

Abstract

We consider a stochastic model where a population grows in batches according to renewal arrival process. The population is prone to be affected by catastrophes which occur according to Poisson process. The catastrophe starts the destruction of the population sequentially, with one individual at a time, with probability p. This process comes to an end when the first individual survives or when the entire population is eliminated. Using supplementary variable and difference equation method we obtain explicit expressions of population size distribution in steady-state at pre-arrival and arbitrary epochs, in terms of roots of the associated characteristic equation. Besides, we prove that the distribution at pre-arrival epoch is asymptotically geometric. Based on our theoretical work we present few numerical results to demonstrate the efficiency of our methodology. We also investigate the impact of different parameters on the behavior of the model through some numerical examples.

Suggested Citation

  • F. P. Barbhuiya & Nitin Kumar & U. C. Gupta, 2019. "Batch Renewal Arrival Process Subject to Geometric Catastrophes," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 69-83, March.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:1:d:10.1007_s11009-018-9643-2
    DOI: 10.1007/s11009-018-9643-2
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    References listed on IDEAS

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    1. Economou, Antonis, 2003. "On the control of a compound immigration process through total catastrophes," European Journal of Operational Research, Elsevier, vol. 147(3), pages 522-529, June.
    2. Spiros Dimou & Antonis Economou, 2013. "The Single Server Queue with Catastrophes and Geometric Reneging," Methodology and Computing in Applied Probability, Springer, vol. 15(3), pages 595-621, September.
    3. Economou, Antonis & Fakinos, Demetrios, 2003. "A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes," European Journal of Operational Research, Elsevier, vol. 149(3), pages 625-640, September.
    4. Epaminondas G. Kyriakidis & Theodosis D. Dimitrakos, 2005. "Computation of the Optimal Policy for the Control of a Compound Immigration Process through Total Catastrophes," Methodology and Computing in Applied Probability, Springer, vol. 7(1), pages 97-118, March.
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    Cited by:

    1. Nitin Kumar & U. C. Gupta, 2020. "A Renewal Generated Geometric Catastrophe Model with Discrete-Time Markovian Arrival Process," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1293-1324, September.
    2. Nitin Kumar & Umesh Chandra Gupta, 2022. "Markovian Arrival Process Subject to Renewal Generated Binomial Catastrophes," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2287-2312, December.
    3. Nitin Kumar & F. P. Barbhuiya & U. C. Gupta, 2020. "Unified killing mechanism in a single server queue with renewal input," OPSEARCH, Springer;Operational Research Society of India, vol. 57(1), pages 246-259, March.

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